To solve the given optimization problem, we first need to convert the natural language description into a symbolic representation. This involves defining variables and formulating the objective function and constraints using these variables.

Let's define the variables as follows:
- $x_1$: The number of hours the north factory runs.
- $x_2$: The number of hours the south factory runs.

The objective is to minimize the total cost of running both factories. Given that running the north factory for an hour costs $200 and the south factory for an hour costs $400, the objective function can be written as:
\[ \text{Minimize} \quad 200x_1 + 400x_2 \]

The constraints are based on the production requirements:
- For long coats: $20x_1 + 30x_2 \geq 75$
- For short coats: $15x_1 + 25x_2 \geq 30$
- For mini coats: $10x_1 + 30x_2 \geq 40$

Additionally, since the factories cannot run for a negative number of hours, we have:
- $x_1 \geq 0$
- $x_2 \geq 0$

Now, let's represent this problem in the requested format:

```json
{
  'sym_variables': [('x1', 'hours the north factory runs'), ('x2', 'hours the south factory runs')],
  'objective_function': '200*x1 + 400*x2',
  'constraints': [
    '20*x1 + 30*x2 >= 75',
    '15*x1 + 25*x2 >= 30',
    '10*x1 + 30*x2 >= 40',
    'x1 >= 0',
    'x2 >= 0'
  ]
}
```

To solve this problem using Gurobi in Python, we'll use the following code:

```python
from gurobipy import *

# Create a new model
m = Model("Tom_Designs_Production")

# Define variables
x1 = m.addVar(name='x1', lb=0)  # Hours north factory runs
x2 = m.addVar(name='x2', lb=0)  # Hours south factory runs

# Set the objective function
m.setObjective(200*x1 + 400*x2, GRB.MINIMIZE)

# Add constraints
m.addConstr(20*x1 + 30*x2 >= 75, name='long_coats')
m.addConstr(15*x1 + 25*x2 >= 30, name='short_coats')
m.addConstr(10*x1 + 30*x2 >= 40, name='mini_coats')

# Optimize model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print(f"Optimal solution found. North factory runs for {x1.x} hours, south factory runs for {x2.x} hours.")
    print(f"Total cost: ${200*x1.x + 400*x2.x}")
else:
    print("No optimal solution found")
```